Initial value problems on manifolds around submanifolds (reference)

Question

I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded submanifold. I am interesting in solutions of PDE's (any kind) formulated on $X$ (or a arbitrary small neighborhood of $Y$ in $X$) with initial conditions on $Y$. The solutions should also exist on an arbitrary small neighborhood of $Y$ (maybe smaller that the neighborhood on which the PDE is formulated). I am not interested in a particular PDE, but rather I would like to know if there is a general theory on this kind of problems.

Cheers, Ben

Answer 1

I think you might want to look into the theory of exterior differential systems. Unfortunately, the theory is only in good shape for real-analytic PDE systems, though there are some situations in which some version of a smooth theory is known to have positive and useful results.

Typically, the theory of EDS is used for systems whose initial value problems are 'naturally' posed on higher codimension submanifolds, and this usually means that the systems involved are overdetermined, at least in their given formulation.

For a simple example, the Cauchy-Riemann equations for a holomorphic function $h$ of two complex variables are overdetermined: They are four first-order PDE for 2 functions of four variables, and the 'natural' initial value problem is to specify the real and imaginary parts of $h$ on a 'totally real' 2-dimensional surface $R\subset\mathbb{C}^2$. In order for this IVP to have a good local solution, you typically need $R$ to be a real-analytic surface and the specified real and imaginary parts of $h$ to be real-analytic functions on $R$.

For another example, consider the so-called "Björling problem": Given a space curve $c\subset\mathbb{R}^3$ and a unit normal vector field $n$ along $c$, find a minimal surface $S\subset\mathbb{R}^3$ that contains $c$ and has $n$ as its unit normal vector field along $c$. This can be formulated as a first order PDE system for a surface in $X^5 = \mathbb{R}^3\times S^2$ that contains the curve $\bigl(c(t),n(t)\bigr)$ in $X$ (and where this 'initial condition' already is supposed to satisfy the ODE $n(t)\cdot c'(t) = 0$). Again, you only get a good existence theory when $c$ and $n$ are real-analytic, but then you do get existence and local uniqueness.

There are many applications of exterior differential systems in differential geometry. In fact, the theory was initiated by Élie Cartan (and developed further by Erich Kähler) specifically for applications to geometric problems.

There is a certain amount of 'overhead' needed to develop in the theory, so if you are looking for a tool to use on a particular problem, you might want to consult with someone who already knows the theory to determine whether EDS is likely to be of use to you for the class of problems you have in mind before you invest in learning the subject (unless, of course, you are just curious about the theory in general).